Chapter 1: Problem 3
Find the limit. $$ \lim _{x \rightarrow 2} x^{4} $$
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on \(x\) in the denominator is greater than \(3 ?\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 0.5 & 0.2 & 0.1 & 0.01 & 0.001 & 0.0001 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ (a) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}\) (b) \(\lim _{x \rightarrow 0^{-}} \frac{x-\sin x}{x^{2}}\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}\) (d) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}\)
(a) Prove that if \(\lim _{x \rightarrow c}|f(x)|=0,\) then \(\lim _{x \rightarrow c} f(x)=0\). (Note: This is the converse of Exercise \(74 .)\) (b) Prove that if \(\lim _{x \rightarrow c} f(x)=L,\) then \(\lim _{x \rightarrow c}|f(x)|=|L|\). [Hint: Use the inequality \(\|f(x)|-| L\| \leq|f(x)-L| .]\)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \arcsin ^{2} x+\arccos ^{2} x=1 $$
In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow 3} \frac{x-2}{x^{2}} $$
Use the position function \(s(t)=-4.9 t^{2}+150\), which gives the height (in meters) of an object that has fallen from a height of 150 meters. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). Find the velocity of the object when \(t=3\).
What do you think about this solution?
We value your feedback to improve our textbook solutions.
